Multi-plane multi-view approach to project the sphere viewing - PowerPoint PPT Presentation

Multi-plane multi-view approach to project the viewing Multi-plane multi-view approach to project the sphere viewing sphere Introduction Global Map Projections Leonardo K. Sacht Linear Perspective leo-ks@visgraf.impa.br Multi-Plane Perspective Projection Visgraf - IMPA Centering on the objects November 25, 2008 Future Work References Multi-plane multi-view approach to project the viewing sphere

Motivations Multi-plane multi-view approach to project the viewing sphere Planar projections are inexpensive to reproduce, easy to Introduction carry, store and display; Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Motivations Multi-plane multi-view approach to project the viewing sphere Planar projections are inexpensive to reproduce, easy to Introduction carry, store and display; Global Map Projections Integration of inexpensive digital cameras with computers; Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Motivations Multi-plane multi-view approach to project the viewing sphere Planar projections are inexpensive to reproduce, easy to Introduction carry, store and display; Global Map Projections Integration of inexpensive digital cameras with computers; Linear Recent progress in detecting and matching features; Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Motivations Multi-plane multi-view approach to project the viewing sphere Planar projections are inexpensive to reproduce, easy to Introduction carry, store and display; Global Map Projections Integration of inexpensive digital cameras with computers; Linear Recent progress in detecting and matching features; Perspective Multi-Plane Development of good blending techniques. Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

The Viewing Sphere Multi-plane multi-view approach to project the viewing sphere It’s the most natural represenation of a scene. The center Introduction of the sphere is the center of projection. Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

The Viewing Sphere Multi-plane multi-view approach to project the viewing sphere It’s the most natural represenation of a scene. The center Introduction of the sphere is the center of projection. Global Map Projections Each point on the scene is identified with a point on the Linear Perspective viewing sphere, via a normaliztion on the coordinates; Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Equirectangular Format Multi-plane multi-view approach to project the viewing sphere It’s an M × 2 M image that represents the 2 ] × [ − π, π ] ⊆ R 2 ; latitude/longitude domain [ − π 2 , π Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Equirectangular Format Multi-plane multi-view approach to project the viewing sphere It’s an M × 2 M image that represents the 2 ] × [ − π, π ] ⊆ R 2 ; latitude/longitude domain [ − π 2 , π Introduction The horizontal axis varies linearly with the longitude and Global Map Projections the vertical axis varies linearly with the latitude; Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Equirectangular Format Multi-plane multi-view approach to project the viewing sphere It’s an M × 2 M image that represents the 2 ] × [ − π, π ] ⊆ R 2 ; latitude/longitude domain [ − π 2 , π Introduction The horizontal axis varies linearly with the longitude and Global Map Projections the vertical axis varies linearly with the latitude; Linear Perspective Very popular format: you can find thousands of Multi-Plane equirectangular images with different resolutions on Flickr: Perspective Projection www.flickr.com/groups/equirectangular; Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Examples Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Examples Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Examples Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Examples Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Expected result Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Global Map Projections Multi-plane multi-view approach to project the viewing sphere It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at Introduction 100 a.C.; Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Global Map Projections Multi-plane multi-view approach to project the viewing sphere It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at Introduction 100 a.C.; Global Map Projections In [2] the author lists 93 different projections and refers to Linear many more; Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Global Map Projections Multi-plane multi-view approach to project the viewing sphere It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at Introduction 100 a.C.; Global Map Projections In [2] the author lists 93 different projections and refers to Linear many more; Perspective Multi-Plane The global projections are the ones that don’t depend on Perspective Projection the image content. Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Mercator Projection Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Perspective Projection Multi-plane multi-view approach to project the viewing sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Linear Perspective Multi-plane multi-view approach to project the viewing Projects the points of the viewing sphere onto a tangent plane: sphere Introduction Global Map Projections Linear Perspective Multi-Plane Perspective Projection Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere

Math of the perspective Multi-plane multi-view approach to project the viewing sphere Given a point p = ( x , y , z ) ∈ S 2 , with x > 0, consider its geographical coordinates: Introduction Global Map p = ( x , y , z ) = (cos θ cos φ, sin θ cos φ, sin φ ) , Projections Linear Perspective where ( θ, φ ) ∈ ( − π 2 , π 2 ) × ( − π 2 , π 2 ). We want a map from S 2 onto � Multi-Plane x =1 , which is chosen to be the Perspective Projection projective plane. Centering on the objects Future Work References Multi-plane multi-view approach to project the viewing sphere