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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

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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

Leonardo K. Sacht leo-ks@visgraf.impa.br

Visgraf - IMPA

November 25, 2008

Multi-plane multi-view approach to project the viewing sphere

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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

Motivations

Planar projections are inexpensive to reproduce, easy to carry, store and display;

Multi-plane multi-view approach to project the viewing sphere

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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

Motivations

Planar projections are inexpensive to reproduce, easy to carry, store and display; Integration of inexpensive digital cameras with computers;

Multi-plane multi-view approach to project the viewing sphere

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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

Motivations

Planar projections are inexpensive to reproduce, easy to carry, store and display; Integration of inexpensive digital cameras with computers; Recent progress in detecting and matching features;

Multi-plane multi-view approach to project the viewing sphere

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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

Motivations

Planar projections are inexpensive to reproduce, easy to carry, store and display; Integration of inexpensive digital cameras with computers; Recent progress in detecting and matching features; Development of good blending techniques.

Multi-plane multi-view approach to project the viewing sphere

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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

The Viewing Sphere

It’s the most natural represenation of a scene. The center

f the sphere is the center of projection.

Multi-plane multi-view approach to project the viewing sphere

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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

The Viewing Sphere

It’s the most natural represenation of a scene. The center

f the sphere is the center of projection.

Each point on the scene is identified with a point on the viewing sphere, via a normaliztion on the coordinates;

Multi-plane multi-view approach to project the viewing sphere

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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

Equirectangular Format

It’s an M × 2M image that represents the latitude/longitude domain [−π

2 , π 2 ] × [−π, π] ⊆ R2;

Multi-plane multi-view approach to project the viewing sphere

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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

Equirectangular Format

It’s an M × 2M image that represents the latitude/longitude domain [−π

2 , π 2 ] × [−π, π] ⊆ R2;

The horizontal axis varies linearly with the longitude and the vertical axis varies linearly with the latitude;

Multi-plane multi-view approach to project the viewing sphere

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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

Equirectangular Format

It’s an M × 2M image that represents the latitude/longitude domain [−π

2 , π 2 ] × [−π, π] ⊆ R2;

The horizontal axis varies linearly with the longitude and the vertical axis varies linearly with the latitude; Very popular format: you can find thousands of equirectangular images with different resolutions on Flickr: www.flickr.com/groups/equirectangular;

Multi-plane multi-view approach to project the viewing sphere

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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

Examples

Multi-plane multi-view approach to project the viewing sphere

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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

Examples

Multi-plane multi-view approach to project the viewing sphere

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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

Examples

Multi-plane multi-view approach to project the viewing sphere

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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

Examples

Multi-plane multi-view approach to project the viewing sphere

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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

Expected result

Multi-plane multi-view approach to project the viewing sphere

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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

Global Map Projections

It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at 100 a.C.;

Multi-plane multi-view approach to project the viewing sphere

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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

Global Map Projections

It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at 100 a.C.; In [2] the author lists 93 different projections and refers to many more;

Multi-plane multi-view approach to project the viewing sphere

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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

Global Map Projections

It’s a very old and long studied problem: for example, Ptolomy claims the invention of geographic projection at 100 a.C.; In [2] the author lists 93 different projections and refers to many more; The global projections are the ones that don’t depend on the image content.

Multi-plane multi-view approach to project the viewing sphere

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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

Mercator Projection

Multi-plane multi-view approach to project the viewing sphere

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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

Perspective Projection

Multi-plane multi-view approach to project the viewing sphere

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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

Linear Perspective

Projects the points of the viewing sphere onto a tangent plane:

Multi-plane multi-view approach to project the viewing sphere

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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

Math of the perspective

Given a point p = (x, y, z) ∈ S2, with x > 0, consider its geographical coordinates: p = (x, y, z) = (cos θ cos φ, sin θ cos φ, sin φ), where (θ, φ) ∈ (−π

2 , π 2 ) × (−π 2 , π 2 ).

We want a map from S2 onto

x=1, which is chosen to be the

projective plane.

Multi-plane multi-view approach to project the viewing sphere

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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

Math of the perspective

T : S2 →

(x, y, z) →

1, y

x , z x

T(cos θ cos φ, sin θ cos φ, sin φ) =

=

1, sin θ cos φ

cos θ cos φ, sin φ cos θ cos φ

=
1, tan θ, tan φ

cos θ

Multi-plane multi-view approach to project the viewing sphere

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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

Inverse Mapping

Given (ˆ y, ˆ z) ∈

x=1, we want to know what are the

geographic coordinates of the point that is projected on (ˆ y, ˆ z): ˆ y = tan θ ˆ z = tan φ cos θ ⇒ θ = arctan ˆ y φ = arctan(ˆ z cos θ) Thus T −1 :

→ S2(≈ [−π, π] × [−π

2 , π 2 ])

(ˆ y, ˆ z) → (arctan(ˆ y), arctan(ˆ z cos(arctan(ˆ y)))) .

Multi-plane multi-view approach to project the viewing sphere

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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

Working with images...

The user chooses an equirectangular image to be projected and a positive real number M such that the projection

nto [−M, M] × [−M, M] ⊆

x=1 will be calculated.

Multi-plane multi-view approach to project the viewing sphere

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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

Working with images...

The user chooses an equirectangular image to be projected and a positive real number M such that the projection

nto [−M, M] × [−M, M] ⊆

x=1 will be calculated.

The algorithm creats an empty square image B wich is proportional to the input equirectangular image.

Multi-plane multi-view approach to project the viewing sphere

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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

Working with images...

The user chooses an equirectangular image to be projected and a positive real number M such that the projection

nto [−M, M] × [−M, M] ⊆

x=1 will be calculated.

The algorithm creats an empty square image B wich is proportional to the input equirectangular image. For each pixel on B, calculate its corresponding position

n [−M, M] × [−M, M] and (θ, φ), via T −1.

Multi-plane multi-view approach to project the viewing sphere

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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

Working with images...

The user chooses an equirectangular image to be projected and a positive real number M such that the projection

nto [−M, M] × [−M, M] ⊆

x=1 will be calculated.

The algorithm creats an empty square image B wich is proportional to the input equirectangular image. For each pixel on B, calculate its corresponding position

n [−M, M] × [−M, M] and (θ, φ), via T −1.

Calculate what pixel on the equirectangular image has the closest coordinates to (θ, φ) and set its color coordinates to be the coordinates of the pixel on B.

Multi-plane multi-view approach to project the viewing sphere

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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

Example: M = 1

Multi-plane multi-view approach to project the viewing sphere

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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

Example: M = 3

Multi-plane multi-view approach to project the viewing sphere

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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

Example: M = 5

Multi-plane multi-view approach to project the viewing sphere

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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 Perspective Projection

More than one plane of projection is chosen and a limited field of view is projected onto each plane;

Multi-plane multi-view approach to project the viewing sphere

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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 Perspective Projection

More than one plane of projection is chosen and a limited field of view is projected onto each plane; This will introduce discontinuities on the final result.

Multi-plane multi-view approach to project the viewing sphere

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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

How to project onto another plane?

With a simple shift on the latitudes:

Multi-plane multi-view approach to project the viewing sphere

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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

How to project onto another plane?

With a simple shift on the latitudes: Shift the longitudes on the equirectangular image to have the center of the projection plane on the center of the image;

Multi-plane multi-view approach to project the viewing sphere

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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

How to project onto another plane?

With a simple shift on the latitudes: Shift the longitudes on the equirectangular image to have the center of the projection plane on the center of the image; Project in the same way as done before;

Multi-plane multi-view approach to project the viewing sphere

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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

Original Equirectangular image

Multi-plane multi-view approach to project the viewing sphere

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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

60 degrees shifted

Multi-plane multi-view approach to project the viewing sphere

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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

Projected image 60 degrees left

Multi-plane multi-view approach to project the viewing sphere

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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

Projected image 60 degrees right

Multi-plane multi-view approach to project the viewing sphere

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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

Projected image with same center

Multi-plane multi-view approach to project the viewing sphere

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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

Combining the three images

Multi-plane multi-view approach to project the viewing sphere

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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

Centering on the objects

Based on Renaissance paintings;

Multi-plane multi-view approach to project the viewing sphere

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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

Centering on the objects

Based on Renaissance paintings; For each foreground object, project the scene to have the

bject on the center of the projection.

Multi-plane multi-view approach to project the viewing sphere

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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

Constructing the panorama in 4 steps

Obtain a foreground-background segmentation for the equirectangular image and cut out the foreground objects;

Multi-plane multi-view approach to project the viewing sphere

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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

Constructing the panorama in 4 steps

Obtain a foreground-background segmentation for the equirectangular image and cut out the foreground objects; Fill in the holes in the background using some texture propagation or inpainting technique;

Multi-plane multi-view approach to project the viewing sphere

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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

Constructing the panorama in 4 steps

Obtain a foreground-background segmentation for the equirectangular image and cut out the foreground objects; Fill in the holes in the background using some texture propagation or inpainting technique; Build a panorama for the background;

Multi-plane multi-view approach to project the viewing sphere

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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

Constructing the panorama in 4 steps

Obtain a foreground-background segmentation for the equirectangular image and cut out the foreground objects; Fill in the holes in the background using some texture propagation or inpainting technique; Build a panorama for the background; Make a projection centered on each foreground object, cut this object from the projection, scale it and paste over the backgorund panorama.

Multi-plane multi-view approach to project the viewing sphere

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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

Step 1 - Obtaining a foreground-background segmentation

I used Intelligent Scissors implementation from GIMP:

Multi-plane multi-view approach to project the viewing sphere

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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

Step 1 - Obtaining a foreground-background segmentation

I used Intelligent Scissors implementation from GIMP:

Multi-plane multi-view approach to project the viewing sphere

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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

Step 2 - Texture Propagation

I used a MATLAB implementation of the reference [1]:

Multi-plane multi-view approach to project the viewing sphere

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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

Step 2 - Filled Image

Multi-plane multi-view approach to project the viewing sphere

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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

Step 3 - Projecting the background

Multi-plane multi-view approach to project the viewing sphere

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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

Result obtained before...

Multi-plane multi-view approach to project the viewing sphere

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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

Discussion on Step 4

In the case of the previous image, it won’t be natural to adopt the idea proposed on the article, because the

bjects (the posters) have a commitment with with the

background (the walls);

Multi-plane multi-view approach to project the viewing sphere

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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

Discussion on Step 4

In the case of the previous image, it won’t be natural to adopt the idea proposed on the article, because the

bjects (the posters) have a commitment with with the

background (the walls); For this case, we just cut the objects from the three images that compose the panorama, scale them and repaste on the projection, reducing distortions, but maintaining the appearence of a prespective projection.

Multi-plane multi-view approach to project the viewing sphere

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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

The result (field of view of about 200 degrees horizontally)

Multi-plane multi-view approach to project the viewing sphere

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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

Discussion on Step 4

People usually don’t have a commitment with the background and their distortions are very appearent even for small fields of view.

Multi-plane multi-view approach to project the viewing sphere

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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

Discussion on Step 4

People usually don’t have a commitment with the background and their distortions are very appearent even for small fields of view. For this case, we’ll adopt what was proposed by the article: For each foreground object, cut it from a projection centered on it and paste it rescaled to have the same height as it were projected with the background.

Multi-plane multi-view approach to project the viewing sphere

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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

Example (with just one projection)

Multi-plane multi-view approach to project the viewing sphere

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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

Example (with just one projection)

Multi-plane multi-view approach to project the viewing sphere

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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

Multiple projections with multiple views

Multi-plane multi-view approach to project the viewing sphere

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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

Multiple projections with multiple views

Multi-plane multi-view approach to project the viewing sphere

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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

Future Work

Automatize the segmentation of the viewing sphere using Variational Shape Approximation. Densities will be chosen according the discontinuities and the objects of the scene;

Multi-plane multi-view approach to project the viewing sphere

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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

Future Work

Automatize the segmentation of the viewing sphere using Variational Shape Approximation. Densities will be chosen according the discontinuities and the objects of the scene; Work with OpenGL to have more flexibility with the images; How to unfold non trivial topologies?

Multi-plane multi-view approach to project the viewing sphere

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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

Future Work

Working with panoramic videos;

Multi-plane multi-view approach to project the viewing sphere

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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

Future Work

Working with panoramic videos; Foreground/background segmentation for panoramic videos compression;

Multi-plane multi-view approach to project the viewing sphere

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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

Future Work

Working with panoramic videos; Foreground/background segmentation for panoramic videos compression; Panoramic videos with different points of view (moving camera);

Multi-plane multi-view approach to project the viewing sphere

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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

Conclusions

Panoramas are of interest of many people, since amateur photographers till scientists;

Multi-plane multi-view approach to project the viewing sphere

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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

Conclusions

Panoramas are of interest of many people, since amateur photographers till scientists; The technique described by the article concludes its goal

f reducing distortions for large field of views, but it’s still

very manual and doesn’t work very well for non man made enviroments;

Multi-plane multi-view approach to project the viewing sphere

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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

Conclusions

Panoramas are of interest of many people, since amateur photographers till scientists; The technique described by the article concludes its goal

f reducing distortions for large field of views, but it’s still

very manual and doesn’t work very well for non man made enviroments; There’s lot of space to work with panoramas...

Multi-plane multi-view approach to project the viewing sphere

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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

References

A. Crismini, P. Prez, K. Toyama: Object Removal by

Exemplar-Based Inpainting. In IEEE Computer Vision and Pattern Recognition, 2003. J.P. Snyder, P.M. Voxland: An Album of Map Projections, Professional Paper 1453. U.S. Geological Survey, 1989.

L. Zelnik-Manor, G. Peters, P. Perona: Squaring the Circle

in Panorams. In Tenth IEEE International Conference in Computer Vision, 2005. Flickr: Equirectangular Group. http://www.flickr.com/groups/equirectangular/

Multi-plane multi-view approach to project the viewing sphere

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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

Thanks!

Coming soon: http : //www.impa.br/ ∼ leo-ks/image processing

Multi-plane multi-view approach to project the viewing sphere