Content- Based Projections for Content-Based Projections for Panoramic Images and Panoramic Images and Videos Videos Introduction Panoramic Leonardo Koller Sacht Images Optimizing Paulo Cezar Carvalho (advisor) Content- Preserving... Luiz Velho (co-advisor) Results Feature Visgraf - IMPA Detection Panoramic Videos April 5, 2010 Conclusion Content-Based Projections for Panoramic Images and Videos
Motivation Content- Based Projections for Panoramic Images and Common cameras capture just a limited field of view of Videos the scene, while our eyes see a much wider field of view Introduction with no obvious distortion; Panoramic Representation of a scene; Images Optimizing Extrapolation of the human perception perception; Content- Preserving... These motivations are even more clear for videos Results (interesting applications); Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Goals Content- Based Projections for Panoramic Images and Videos Study and understand the problem of finding acceptable panoramic images; Introduction Panoramic Deeply detail one reference on this topic; Images Propose extensions ; Optimizing Content- Preserving... Focus on mathematical aspects of the problem; Results Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Pipeline Content- Based Projections for Panoramic Images and A time line: Videos Introduction Panoramic Images Optimizing Content- Preserving... Figure: Structure of the presentation. Results Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
The viewing sphere Each point has an associated color, the color that is seen when Content- Based one looks toward this point. Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Figure: A viewing sphere (looked from outside) that represents the Conclusion visible information of some scene. Content-Based Projections for Panoramic Images and Videos
Longitude/latitude representation Content- Based � � Projections − π 2 , π S 2 r : [ − π, π ] × → for 2 Panoramic ( λ, φ ) �→ (cos( λ ) cos( φ ) , sin( λ ) cos( φ ) , sin( φ )) Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Conclusion Figure: Longitude/latitude representation r . Content-Based Projections for Panoramic Images and Videos
Equirectangular Images Content- Based Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Figure: “San Marco Plaza”, by Flickr user Veneboer, taken from [1]. Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Equirectangular Images Content- Based Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Figure: “Reboot 8.0: Ianus demos Cabinet to Thomas’ kid”, by Panoramic Videos Flickr user Aldo, taken from [1]. Conclusion Content-Based Projections for Panoramic Images and Videos
Problem Statement Content- Based Projections for Panoramic Images and Videos We formulate the panoramic image problem as the one of finding a projection Introduction Panoramic S ⊆ S 2 R 2 u : → Images ( u , v ) , ( λ, φ ) �→ Optimizing Content- Preserving... with desirable properties. Results Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Perspective Projection � � � � − π 2 , π − π 2 , π R 2 P : × → Content- � � 2 2 Based tan( λ ) , tan( φ ) ( λ, φ ) �→ ( u , v ) = Projections cos( λ ) for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Conclusion Figure: Left: 90 degree long./90 degree lat.; Right: 130/120. Content-Based Projections for Panoramic Images and Videos
Stereographic Projection � � − π 2 , π R 2 S : ( − π, π ) × → Content- � � 2 Based 2 sin( λ ) cos( φ ) 2 sin( φ ) ( λ, φ ) �→ cos( λ ) cos( φ )+1 , Projections cos( λ ) cos( φ )+1 for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Conclusion Figure: Left: 180 degree long./180 degree lat.; Right: 180/180. Content-Based Projections for Panoramic Images and Videos
Mercator Projection � � − π 2 , π R 2 M : [ − π, π ) × → Content- 2 Based ( λ, φ ) �→ ( u , v ) = ( λ, log(sec( φ ) + tan( φ ))) Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Conclusion Figure: 360 degree longitude/150 degree latitude. Content-Based Projections for Panoramic Images and Videos
Correction of Geometric Perceptual Distortions in Pictures (Zorin et. al, [2]) An optimization solution is proposed to compromise between Content- Based preservation of lines and shapes of objects: Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Conclusion Figure: Correction with λ = 1 2 . Content-Based Projections for Panoramic Images and Videos
Squaring the Circle in Panoramas (Zelnik-Manor et. al, [3]) Content- Different perspective projections are combined in a way that Based fits the geometry of the scene Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Figure: 180 degree longitude/90 degree longitude. Conclusion Content-Based Projections for Panoramic Images and Videos
Optimizing Content-Preserving Projections for Wide-Angle Images (Carroll et. al, [4]) Content- Based Projections for Panoramic Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Figure: A (cropped) result produced by the method described in this Videos chapter. FOV: 285 degree longitude/170 degree latitude. Conclusion Content-Based Projections for Panoramic Images and Videos
Desirable properties Content- Based Projections for Panoramic Images and Preserve straight lines; Videos Preserve shape of objects; Introduction Vary scale and orientation smoothly; Panoramic Images Depend on the scene content (without being restricted to Optimizing scenes with particular structure); Content- Preserving... Mathematically formalize distortions; Results Handle wide fields of view... Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Discretization of the Viewing Sphere Content- λ ij = − π + j 2 π n , φ ij = − π 2 + i π Based m , j = 0 , . . . , n , i = Projections for 0 , . . . m , Panoramic Images and u ij = u ( λ ij , φ ij ) = ( u ij , v ij ) , j = 0 , . . . , n , i = 0 , . . . m . Videos Introduction Panoramic Images Optimizing Content- Preserving... Results Feature Detection Panoramic Videos Figure: Discretization of the equirectangular domain. Conclusion Content-Based Projections for Panoramic Images and Videos
Preservation of Shapes ↔ Conformality Content- Based Projections Definition 2.1: A dipheomorfism ϕ : S → S is a conformal for Panoramic mapping if for all p ∈ S and for all v 1 , v 2 ∈ T p S holds Images and Videos � d ϕ p ( v 1 ) , d ϕ p ( v 2 ) � = Θ 2 ( p ) � v 1 , v 2 � , Introduction where Θ 2 is a differentiable function on S that never vanishes. Panoramic Images Optimizing Content- Preserving... Locally the mapping preserves angles and inner products Results (except for the stretching factor Θ 2 ( p )). Feature Detection Panoramic In our case, S = S 2 , S = R 2 and ϕ = u . Videos Conclusion Content-Based Projections for Panoramic Images and Videos
Differential vectors Content- Based Projections Differential north vector: for � � � ∂ u � Panoramic � 0 � � Images and ∂φ ( p ) ∂ r Videos h = d u p ∂φ ( p ) = d u p = ∂ v 1 ∂φ ( p ) Introduction Differential east vector: Panoramic Images � � � � � ∂ u � � 1 Optimizing ∂ r 1 ∂λ ( p ) Content- cos( φ ) k = d u p ∂λ ( p ) = d u p = Preserving... ∂ v 0 cos( φ ) ∂λ ( p ) Results Feature Where � ∂ u � Detection ∂ u ∂λ ( p ) ∂φ ( p ) Panoramic d u p = . Videos ∂ v ∂ v ∂λ ( p ) ∂φ ( p ) Conclusion Content-Based Projections for Panoramic Images and Videos
Differential vectors Content- Based Projections for Panoramic Lemma 2.1: u conformal ⇔ h = R 90 k or h = R − 90 k . Images and Videos Introduction Panoramic Images Optimizing Content- Preserving... Figure: We exclude the second possibility above. Results Feature Detection Panoramic Videos Conclusion Content-Based Projections for Panoramic Images and Videos
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