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Correction of Geometric Perceptual Distortions in Pictures 1995

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Correction of Geometric Perceptual Distortions in Pictures 1995 Correction of Geometric Perceptual Distortions in Pictures Thesis by Denis N Zorin In Partial Fulllment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena California i Abstract We suggest an approach for correcting several typ es of p erceived geometric distortions in computer generated and photographic images The approach is based on a mathematical formalization of desirable prop erties of pictures Weprovide a review of p erception of pictures and identify p erceptually imp ortant geometric prop erties of images From a small set of simple assumptions we obtain p erceptually preferable pro jections of three dimensional space into the plane and show that these pro jections can b e decomp osed into a p er sp ective or parallel pro jection followed by a planar transformation The decomp osition is easily implemented and provides a convenient framework for further analysis of the image mapping Weprove that two p erceptually imp ortant prop erties are incompatible and cannot b e satised simultaneously It is imp ossible to construct a pro jection such that the images of all lines are straight and the images of all spheres are exact circles Perceptually preferable tradeos b etween these two typ es of distortions can dep end on the content of the picture We construct parametric families of pro jections with parameters representing the relative imp ortance of the p erceptual characteristics By adjusting the settings of the parameters we can minimize the overall distortion of the picture It turns out that a simple family of transformations pro duces results that are suciently close to optimal We implement the prop osed transformations and apply them to computergenerated and photographic p ersp ective pro jection images Our transformations can considerably reduce distortion in wideangle motion pictures and computergenerated animations ii Contents Abstract i Intro duction Motivation Contributions Persp ective Pro jection and Viewing Position Nomenclature Summary Perception of Pictures Assumptions and Denitions Perceptual Nomenclature Dening Pictures Assumptions ab out Perception Robustness of Pictures Perception of Ob jects in Pictures Rectangular Ob jects Spheres Cylinders Humans and Animals Foreshortening Other Prop erties of Visual Perception Visual Field Perception of Straightness and Curvature Perception of Verticality Linear Persp ective Contents iii Rectangular Ob jects in Linear Persp ective Spheres Cylinders Humans Straightness VerticalityTexture DensityMovement Other Traditional Persp ective Systems Summary Formalization of Perceptual Requirements Technical Requirements Structural Requirements Denitions and Preliminary Lemmas Holes in Images Lo ops Folds and Twists Lo ops and Twists Decomp osition Nonstructural Requirements Denitions Zero Curvature Condition Direct View Condition Error Functionals Optimization Problem Construction of Pro jections Previous Work Perceptuallybased systems Artistic systems Technical Pro jections Axially Symmetric Pro jection Reformulation of the Problem Limiting Cases Lower Bound of the Optimal Values of F Second Bound Contents iv Interp olations Extentions Implementation and Applications Implementation Deep and Shallow Scenes When Do We Need a WideAngle Pro jection Applications of Correcting Transformations Other Applications Summary Conclusions and Future Work Conclusions Future Work App endices A Review of the Exp erimental Studies of Perception of Pictures A Robustness and Perception of the Picture Surface and Frame A Rigidity in Motion Pictures A Size and Distance in Pictures A Orientation and Layout A Rectangular Corners A Foreshortening A Visual eld A Straightness and Curvature A Verticality ter of pro jection from a pro jection of a rectangle B Reconstruction of the cen Bibliography Chapter Intro duction Motivation The pinhole photograph from Pirenne Figure demonstrates the main problem that is considered in this thesis The sphere in the picture app ears to b e deformed and only the fact that ellipsoidal architectural decorations are very uncommon in ancient buildings makes us assume that the ob ject in the picture is indeed a sphere If we strip the picture of additional cues we are likely to p erceive the ob ject in the picture as an ellipsoidlike shap e rather than a sphere The shap e of twodimensional images in a photograph is determined by the pro jection of three dimensional space into the plane pro duced by the camera For a pinhole camera and for most standard lenses this pro jection is the perspective projection The distortion in Figure is one of several typ es of distortion that are quite common for wideangle p ersp ective images There are several questions that we can ask ab out these distortions Why do es p ersp ective pro jection result in distorted images Whydowe p erceive p ersp ective pro jection images as realistic in most cases Howdowe obtain a picture similar to Figure which lo oks less distorted The p ossibility of decreasing destortion is demonstrated in Figure These questions are imp ortant for image synthesis b ecause most existing rendering systems Contributions Figure Left Wideangle pinhole photograph taken on the ro of of the Church of St Ignatzio in Rome classical example of p ersp ective distortions from Pirenne Figure Right Corrected version of the picture with transformation describ ed in Section applied mo del some asp ects of photographic pro cess practically all of them use p ersp ective pro jection as the viewing transformation Answering the questions that wehave p osed requires analysis of some features of human p er ception of pictures for example the p erception of shap e Such analysis can b e used to evaluate commonly used pro jections and create new ones The main goals of the work describ ed in this thesis were to collect and systematize available data on the p erception of pictures of threedimensional scenes todevelop a framework for evaluation and construction of viewing transformation based on p er ceptual considerations to construct sample families of transformations in this framework Contributions We formalize a numb er of p erceptual requirements as mathematical constraints imp osed on the viewing transformation Weprove that a viewing transformation can pro duce p erceptually acceptable images for arbi Viewing transformation is the computer graphics term for the pro jection of threedimensional space into the plane Persp ective Projection and Viewing Position Nomenclature principal ray projection(picture) projection(picture) plane center plane of projection picture picture α α center of center of the picture the picture projection angle direction of projection direction of projection Figure Left Central pro jection Figure Right Parallel pro jection trary scene geometry only if it can b e represented as a comp osition of a linear pro jection and planar transformation for each disjoint part of the scene Weintro duce quantitative measures for twotyp es of p erceptual distortion that can b e used for evaluation of viewing transformations We demonstrate that two imp ortanttyp es of distortion cannot b e eliminated simultaneously We construct a useful family of axially symmetric viewing transformations whichallow the reduction p erceptual distortion We describ e a nonsymmetric extension of this family which allows further reduction of geo metric distortion Persp ective Projection and Viewing Position Nomenclature We shall use a numb er of terms describing various characteristics of p ersp ective pro jections and viewing p ositions There is a lot of variation in usage of some of these terms we try to use a consistent nomenclature whenever p ossible even when describing results of authors who originally used a dierentoneTerminology more sp ecic to perception can b e found in Section and denitions related to more general typ es of pro jection are given in Section and Section central pro jection Let C be a pointinR letp b e a plane such that C pFor anypoint l to b e the line through C and xThenacentral projection with the x R x C dene x center C to the plane p is the mapping R n C p which maps anypoint x R to l p x Persp ective Projection and Viewing Position Nomenclature parallel pro jection Let p a plane v avector which is not parallel to the plane pFor any p oint x R dene l to b e the line through x parallel to v Then a paral lel projection in the x direction v to the plane p is the mapping R p wqhich maps any p oint x R to l p x perspective pro jection Either central or paral lel projection of a part of the threedimensional space into a plane the projection or picture plane orthogonal pro jection A p ersp ective pro jection is orthogonal if the direction of pro jection is p erep endicular to the plane of pro jection this denition applies
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