12/15/2006 15-862 Fall 2006

Two-Point Perspective Single View Geometry

15-862 Final ProjectFaustinus Kevin Gozali (fkg@andrew.cmu.edu)

an extended tour into the picture…

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Two-point Perspective World Camera/eye at (0,0,h)

h = dist(horizon, bottom of image) Camera vanishes at the center of horizon (veye) Image plane parallel to x and z axis

Center at (0,f,h) Two vanishing points (vpl, vpr)

Xvprvpl

veye

Image Plane

x

z

y

yy Bottomhorizonh

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Our Assumptions Horizontal Horizon line

This holds for two-point perspective 1 VP to the left, 1 to the right

Baselines below horizon Uniform heights for all walls

Xvprvpl

veye

Image Plane

x

z

ybase line

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Vanishing Point Calculation User specifies parallel lines Compute intersection point with Least Square Method = VP

Method described by Bob Collins http://www.cs.cmu.edu/~ph/869/www/notes/vanishing.txt

Simply doing MLdivide doesn’t work Done separately for VPleft and VPright

Xvprvpl

veye

Image Plane

x

z

y

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The Horizon Line Calculated based on 2 VPs Assumption: horizontal horizon in image plane Or, user can specify directly

2.. yryl

y

VPVPhorizon

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Selecting Base Lines User specifies base-lines

For each desired vertical wall Assumption: must be below horizon

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Computing Depth (y-axis) Similar to 1-point perspective Based on height ratios

Camera height as reference Recall: image plane at y = f

veye

Image Plane

y

zSide View

dyh

base point depth (3Dy)f

Camera (0,0,h) horizon line fdyhDy 3

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Computing Location in x-axis Using calculated depths (y) Compute horizontal distance from veye in image plane

Recall: left side is x-, right side is x+ Horizontal distance amplified using depth ratio to

camera height Farther points are amplified more

fD

vDD yxeyexx

3)2(3 .X

veye

Image Plane

dx2

dx1

2Dx1

2Dx2

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Specifying Height After 3D location is computed Compute Height Scales

Ratio for each base point to camera height Extend walls vertically upward

Height of the base-point closest to the camera as reference User picks the desired height, then compute 3Dheight

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Ground Surface Ground surface is a prefect rectangle at z = 0 Create Ground mask

Up to the farthest base-point Warp texture (using Homography)

Point correspondences have to consider 3D depth! Create 3D surface

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Vertical Walls Similar to ground processing

All walls are perfect rectangle No mask is needed Warp texture using Homography

Consider distance of each base point as width Uniform height

Create 3D model Intersects the ground plane correctly

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3D Model Generation Define the surfaces

Based on (3Dx, 3Dy, 3Dheight) Wrap textures Observe model

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Fun with Texture Sources Texture Interpolation

approximately evening view Blending

Half morning half night Use this for 3D model

))(1()( NightViewwMorningVietexture

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Texture Examples

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Gallery

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Gallery

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Gallery

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Gallery

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References[1] Chu, Siu-Hang, Animating Chinese Landscape Paintings and Panoramas. A Thesis

Submitted to the Hong Kong University of Science and Technology, August 2001.

[2] Hoeim, Derek; Efros, A. Alexei; Herbert, Martial. Automatic Photo Pop-up. Robotics Institute, Carnegie Melon University, Pittsburgh PA, USA. http://www.cs.cmu.edu/~dhoiem/projects/popup

[3] Single View Reconstruction Lecture slides. http://graphics.cs.cmu.edu/courses/15-463/2006_fall/www/Lectures/SingleViewReconstruction.pdf

[4] Perspective Drawing. An online tutorial. http://www.lems.brown.edu/vision/people/leymarie/SkiP/May98/Boehm1.html

[5] Horry, Yoichi; Anjyo, Ken-ichi; Arai, Kiyoshi. Tour Into the Picture: Using a Spidery Mesh Interface to Make Animation from a Single Image. Hitachi, Ltd.

[6] Collins, Bob. A guide to compute vanishing points. http://www.cs.cmu.edu/~ph/869/www/notes/vanishing.txt

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Q&A

Questions?