Quadric Transfer for Immersive Curved Screen Displays

1
Quadric Transfer for Immersive Curved Screen
Displays

• Ramesh Raskar, Jeroen van Bar, Thomas
  Willwacher, Srinivas Rao
• Mitsubishi Electric Research Labs
• Cambridge, MA USA

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Curved Screen Displays

• Multiple overlapping projectors
• Goal
• Replace single-proj
• Higher resoltn
• Higher brightness
• Sub-pixel auto-alignment
• Parametric solution

Planetarium
Simulators
3
Dome Projection
Spherical Warping

• Edge-Blended
• Display

Sub-Frames
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(No Transcript)
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OutlineSeamless Curved Display

• Multi-projector methods
• Simplified Quadric Image Transfer
• Calibration with camera-pair
• Parametric Rendering solution

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OutlineSeamless Curved Display

• Multi-projector methods
• Simplified Quadric Image Transfer
• Calibration with camera-pair
• Parametric Rendering solution

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Multi-Projector Displays
Precise config Costly setup Manual alignment
High maintenance
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Multi-Projector Displays
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Multi-Projector Displays
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Multi-Projector Displays
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Planar Displaywith parametric approach
R Raskar
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Planar projective transfer homography

• Two images of 3D points on a plane
• Related by a 3x3 matrix

x A3 x 3 x
M
x
A3 x 3
x
x
x
Proj 1
Proj 2
Proj 1
Proj 2
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Parametric Image Transfer
X
x
x
Planar Homography
Quadric Transfer
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Parametric Approach

• Calibration
• Lower camera resolution
• Tolerance for pixel localization errors
• Faster calibration
• Rendering
• Efficient well-defined warping
• Avoids look up tables

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Multi-Projector Displays
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Curved ScreensView for a Sweet-spot
Projector
Sweet spot
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Calibration for a Sweet-spot
Projector
Camera at Sweet spot
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Discretized non-parametric approach
Projector Image
p1
p6
c1
c6
Camera Image
Desired Image

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Pre-Warped Projection

• Discretized Warping Software
• Spitz - PolyDome

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OutlineSeamless Curved Display

• Multi-projector methods
• Simplified Quadric Image Transfer
• Calibration with camera-pair
• Parametric Rendering solution

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Quadric classification
Projectively equivalent to sphere
sphere
ellipsoid
paraboloid
hyperboloid
Ruled quadrics
hyperboloids of one sheet
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Quadrics
4x4 symmetric matrix,
Nine d.o.f
X
Q
9 points in 3D define quadric
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Quadric Image Transfer
Shashua97
X
If
,
x
x
21 params, 4 more than necessary !
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Simplified Quadric Image Transfer
Based on..
X
Homography with polar plane
Projected conic
x
x
17 param warp
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Simplified Quadric Image Transfer
X
17 param warp
x
x
Planar homography 4 corresponding
pixels Quadric transfer 9 corresponding pixels
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OutlineSeamless Curved Display

• Multi-projector low cost method
• Simplified Quadric Image Transfer
• Calibration with camera-pair
• Parametric Rendering solution

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Calibration of Quadric Screens
1
Dome Screen
2
4
3
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Approach

• Calibration
• At each projector i ,
• Project structured pattern
• View with stereo camera
• Finding camera to projector quadric transfer,
• Run-time
• At each projector i,
• Pre-warp input image using

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CalibrationFinding relationship between camera
and projector
Low-res Camera 640x480 images But each Projector
1024x768
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Before Blending
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After Blending
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Intensity Correction in Overlap
Projector Framebuffers
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Projector Framebuffers
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Projector Framebuffer Intensity Weights
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OutlineSeamless Curved Display

• Multi-projector low cost method
• Simplified Quadric Image Transfer
• Calibration with camera-pair
• Parametric Rendering solution

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Rendering a 3D Scene

• Steps at each projector
• (Pre-distort vertex 3D location)
• For each triangle T with vertices Mj
• For each vertex M
• Find pixel m via VirtualViewProjection( M )
• Find warped pixel m via quadricTransfer of m
• Replace M with m

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Vertex Shader for Quadric Transfer in Cg

• vertout main( appin IN, uniform float4x4
  modelViewProj, uniform float4 constColor,
  uniform float3x3 A, uniform float3x3 E,
  uniform float3 e)
• vertout OUT
• float4 m1 float4(IN.position.x, IN.position.y,
  IN.position.z, 1.0f )
• float4 m, mi float3 m2,mp float scale
•  
• m mul( modelViewProj, m1)
• m2.x m.x/m.w m2.y m.y/m.w m2.z 1
• scale mul(m2, mul(E,m2))
• mp mul(A,m2) sqrt(scale)e
• mi.x m.w (mp.x)/(mp.z)
• mi.y m.w (mp.y)/(mp.z)
• mi.zw m.zw
• OUT.position mi 
• OUT.color0 IN.color0 // Use the original
  per-vertex color specified
• return OUT

ParametricWarp
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3D Terrain Rendering
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Parametric Rendering Benefits

• Head tracking support
• Update quadric transfer per frame
• Single pass rendering
• Avoid post-rasterized warp
• Efficient rendering
• Better image quality
• Programmable hardware
• Distributed rendering
• Runs unmodified 3D applications

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Head Tracked Single Pass Rendering
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Distributed Rendering with Unmodified Application
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Subpixel Accurate Registration
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Convex Dome
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Acknowledgements

• Mitsubishi Electric Research Labs
• Paul Beardsley, Jay Thornton
• Joe Marks
• Mitsubishi Electric, Japan
• Masato Ogata, Hiroyuki Wada
• Masatoshi Kameyama, Ashizaki

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Seamless Curved Display

• Multi-projector low cost method
• Simplified Quadric Image Transfer
• Complete Parametric calibrendering solution
• Head tracking support, single pass rendering

www.MERL.com/Projects/Projector/
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Details I Skipped ..

• Photometric Correction Majumder03
• Depth of field is limited
• Estimating camera and projector params
• Internal and External params
• Issue with near-planar 3D points
• Finding pixels weights for blending
• Non-linear optimization
• Rendering
• Warping and Depth buffer issues

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Advantages

• Parametric warp
• Lower camera resolution
• Tolerance for pixel localization errors
• Faster calibration
• Efficient well-defined warping

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What is homography ?

j A3 x 3 i
A3 x 3
a1 a2 a3 b1 b2 b3 c1 c2 c3
jx jy 1
ix iy 1

k
j
i

jx (a i) / (c i)
Proj 1
Proj 2

jy (b i) / (c i)
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Rendering 2D 3D scene
Concave Dome
Convex Dome
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Projector Mailing List

• majordomo_at_cs.unc.edusubscribe projector
• Projector bibliography

www.raskar.com/Projector/
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Vertex Shader for Quadric Transfer in Cg

• vertout main( appin IN, uniform float4x4
  modelViewProj, uniform float4 constColor,
  uniform float3x3 A, uniform float3x3 E,
  uniform float3 e)
• vertout OUT
• float4 m1 float4(IN.position.x, IN.position.y,
  IN.position.z, 1.0f )
• float4 m, mi float3 m2,mp float scale
•  
• m mul( modelViewProj, m1)
• m2.x m.x/m.w m2.y m.y/m.w m2.z 1
• scale mul(m2, mul(E,m2))
• mp mul(A,m2) sqrt(scale)e
• mi.x m.w (mp.x)/(mp.z)
• mi.y m.w (mp.y)/(mp.z)
• mi.zw m.zw
• OUT.position mi 
• OUT.color0 IN.color0 // Use the original
  per-vertex color specified
• return OUT

ParametricWarp
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Planar Displays
Current Multi-Cube System
MERL Projector Planar Mosaic
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Offset Sweetspot
Ideal Sweetspot
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Quadric Transfer

• Convex or concave quadric surfaces
• Minutes rather than seconds due to necessary
  non-linear optimization of quadric transfer
  parameters
• Sub-pixel accuracy
• Current consumer-class graphics hardware allows
  fast rendering

Concave
Convex
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Non-linear Refinement
Linear Estimation Error 10 pixels
NonLinear Refinement Error 1.0 pixels
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Traditional Multi-Projector Setup
Well-defined Overlap
No Overlap
Precise configuration Costly setup Manual
alignment High maintenance
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Related Work

• Conventional Displays
• Manual alignment, expensive infrastructure
• Jupiter,Trimensions, CAVE, Planetaria, Flight
  Simulators
• Planar Screens
• Camera in loop, auto calibration, low cost
• Exploit homography parameters
• Raskar98,Surati99,Chen00,Brown02
• Curved Screens
• Non-parametric solutions
• Jarvis97,Raskar98,Yang01
• Parametric
• ?, Siggraph 2003