Title: A SignalProcessing Framework for Forward and Inverse Rendering
1A Signal-Processing Framework for Forward and
Inverse Rendering
Ph.D. Orals June 3, 2002
2Illumination Illusion
- People perceive materials more easily under
natural illumination than simplified
illumination.
Images courtesy Ron Dror and Ted Adelson
3Illumination Illusion
- People perceive materials more easily under
natural illumination than simplified
illumination.
Images courtesy Ron Dror and Ted Adelson
4Material Recognition
Photographs of 4 spheres in 3 different lighting
conditions courtesy Dror and Adelson
5Estimating BRDF and Lighting
Photographs
Geometric model
6Estimating BRDF and Lighting
Forward RenderingAlgorithm
Photographs
BRDF
Rendering
Lighting
Geometric model
7Estimating BRDF and Lighting
Forward RenderingAlgorithm
Photographs
BRDF
Novel lighting
Rendering
Geometric model
8Inverse Problems Difficulties
Ill-posed (ambiguous)
9Real-Time Rendering
- Interactive rendering with natural lighting,
physical BRDFs
10Contributions
- Formalize reflection as convolution
- Signal-processing framework
- Practical forward and inverse algorithms
11Outline
- Motivation
- Reflection as Convolution
- Preliminaries, assumptions
- Reflection equation, Fourier analysis (2D)
- Spherical Harmonic Analysis (3D)
- Signal-Processing Framework
- Applications
- Summary and Implications
12Assumptions
13Assumptions
- Known geometry
- Convex curved surfaces no shadows,
interreflection
Complex geometry use surface normal
14Assumptions
- Known geometry
- Convex curved surfaces no shadows,
interreflection - Distant illumination
Illumination Grace Cathedral courtesy Paul
Debevec
Photograph of mirror sphere
15Assumptions
- Known geometry
- Convex curved surfaces no shadows,
interreflection - Distant illumination
- Homogeneous isotropic materials
Anisotropic
Isotropic
16Assumptions
- Known geometry
- Convex curved surfaces no shadows,
interreflection - Distant illumination
- Homogeneous isotropic materials
- Later, practical algorithms relax some
assumptions
17Reflection
18Reflection as Convolution (2D)
L
B
19Reflection as Convolution (2D)
20Reflection as Convolution (2D)
21Reflection as Convolution (2D)
Fourier analysis
R. Ramamoorthi and P. Hanrahan Analysis of
Planar Light Fields from Homogeneous Convex
Curved Surfaces under Distant Illumination SPIE
Photonics West 2001 Human Vision and Electronic
Imaging VI pp 195-208
22Related Work
- Qualitative observation of reflection as
convolution Miller Hoffman 84, Greene
86, Cabral et al. 87,99 - Reflection as frequency-space operator DZmura
91 - Lambertian reflection is convolution Basri
Jacobs 01 - Our Contributions
- Explicitly derive frequency-space convolution
formula - Formal quantitative analysis in general 3D case
23Spherical Harmonics
0
1
2 . . .
-1
-2
0
1
2
24Spherical Harmonic Analysis
2D
3D
25Outline
- Motivation
- Reflection as Convolution
- Signal-Processing Framework
- Insights, examples
- Well-posedness of inverse problems
- Applications
- Summary and Implications
26Insights Signal Processing
- Signal processing framework for reflection
- Light is the signal
- BRDF is the filter
- Reflection on a curved surface is convolution
27Insights Signal Processing
- Signal processing framework for reflection
- Light is the signal
- BRDF is the filter
- Reflection on a curved surface is convolution
Filter is Delta function Output Signal
28Insights Signal Processing
- Signal processing framework for reflection
- Light is the signal
- BRDF is the filter
- Reflection on a curved surface is convolution
Signal is Delta function Output Filter
29Phong, Microfacet Models
Mirror
Illumination estimation ill-posed for rough
surfaces
Analytic formulae in R. Ramamoorthi and P.
Hanrahan A Signal-Processing Framework for
Inverse Rendering SIGGRAPH 2001 pp 117-128
30Lambertian
Incident radiance (mirror sphere)
Irradiance (Lambertian)
31Inverse Lighting
Given B,? find L
- Well-posed unless denominator vanishes
- BRDF should contain high frequencies Sharp
highlights - Diffuse reflectors low pass filters Inverse
lighting ill-posed
32Inverse BRDF
Given B,L find ?
- Well-posed unless Llm vanishes
- Lighting should have sharp features (point
sources, edges) - BRDF estimation ill-conditioned for soft lighting
Area source Same BRDF
Directional Source
33Factoring the Light Field
- Light Field can be factored
- Up to global scale factor
- Assumes reciprocity of BRDF
- Can be ill-conditioned
- Analytic formula derived
Given B find L and ?
More knowns (4D) than unknowns (2D/3D)
34Outline
- Motivation
- Reflection as Convolution
- Signal-Processing Framework
- Applications
- Forward rendering (convolution)
- Inverse rendering (deconvolution)
- Summary and Implications
35Computing Irradiance
- Classically, hemispherical integral for each
pixel - Lambertian surface is like low pass filter
- Frequency-space analysis
Incident Radiance
Irradiance
369 Parameter Approximation
Order 0 1 term (constant)
Exact image
RMS error 25
379 Parameter Approximation
Order 1 4 terms (linear)
Exact image
RMS Error 8
389 Parameter Approximation
Order 2 9 terms (quadratic)
Exact image
RMS Error 1
For any illumination, average error lt 2 Basri
Jacobs 01
39Comparison
Irradiance map Texture 256x256 Hemispherical Inte
gration 2Hrs
Irradiance map Texture 256x256 Spherical
Harmonic Coefficients 1sec
Incident illumination 300x300
40Video
R. Ramamoorthi and P. Hanrahan An Efficient
Representation for Irradiance Environment Maps
SIGGRAPH 2001 pp 497-500 R. Ramamoorthi and P.
Hanrahan Frequency Space Environment Map
Rendering SIGGRAPH 2002
41Video
42Inverse Rendering Previous Work
Lighting
Unknown
Known
Known
BRDF
Unknown
Textures are a third axis
43Contributions
- Complex illumination
- Factorization of BRDF, lighting (find both)
- New representations and algorithms
- Formal study of inverse problems (well-posed?)
44Complications
- Incomplete sparse data (few photographs)
- Concavities Self Shadowing
- Spatially varying BRDFs
45Complications
- Challenge Incomplete sparse data (few
photographs) Difficult to compute
frequency spectra - Solution
- Use parametric BRDF model
- Dual angular and frequency space representation
46Algorithm Validation
Photograph
True values
47Algorithm Validation
Photograph
Renderings
Image RMS error 5
Known lighting
Unknown lighting
True values
48Inverse BRDF Spheres
Photographs
Renderings (Recovered BRDF)
49Complications
- Challenge Complex geometry with concavities
Self shadowing - Solution
- Use associativity of convolution
- Blur lighting, treat specular BRDF term as mirror
- Single ray for shadowing, easy in ray tracer
50Complex Geometry
- 3 photographs of a sculpture
- Complex unknown illumination
- Geometry known
- Estimate microfacet BRDF and distant lighting
51Comparison
52New View, Lighting
Photograph
Rendering
53Complications
- Challenge Spatially varying BRDFs
- Solution
- Use textures to modulate BRDF parameters
54Textured Objects
Rendering
Photograph
55Summary
- Reflection as convolution
- Frequency-space analysis gives many insights
- Practical forward and inverse algorithms
- Signal-Processing A useful paradigm for forward
and inverse rendering in graphics and vision
56Implications and Future Work
- Duality between forward and inverse problems
- Analyzing intrinsic structure of light field
- How many images in image-based rendering?
- How many principal components in PCA?
- Differential framework for reflection
- Complex illumination in computer vision
57Papers
- R. Ramamoorthi and P. Hanrahan A
Signal-Processing Framework for Inverse
Rendering SIGGRAPH 2001 pp 117-128 - R. Ramamoorthi and P. Hanrahan An Efficient
Representation for Irradiance Environment Maps
SIGGRAPH 2001 pp 497-500 - R. Ramamoorthi and P. Hanrahan Frequency Space
Environment Map Rendering SIGGRAPH 2002 - R. Ramamoorthi and P. Hanrahan On the
Relationship between Radiance and Irradiance
Determining the Illumination from images of a
Convex Lambertian Object Journal of the Optical
Society of America A 18(10) 2001 pp 2448-2459 - R. Ramamoorthi and P. Hanrahan Analysis of
Planar Light Fields from Homogeneous Convex
Curved Surfaces under Distant Illumination SPIE
Photonics West 2001 Human Vision and Electronic
Imaging VI 195-208
ravir_at_graphics.stanford.edu http//graphics.stanf
ord.edu/ravir
58Acknowledgements
- Pat Hanrahan
- Committee Marc, Jitendra, Ron, Bernd
- Szymon Rusinkiewicz and Steve Marschner
- Bill Mark and Kekoa Proudfoot
- GLAB
- gerth, ada, heather, seander, maneesh,
henrik, jedavis, olaf, humper, dk, renng,
tpurcell, psen, vaibhav, zsh, munzner, lucasp,
liyiwei, - Hodgson-Reed Stanford Graduate Fellowship
- NSF ITR grant 0085864 Interacting with the
Visual World
59The End
60Motivation
- Understand nature of reflection and illumination
- Applications in computer graphics
- Real-time forward rendering
- Inverse rendering
-
61Photorealistic Rendering
62Measuring Materials, Light
Measure BRDF (reflectance) Point light source
63Interactive Forward Rendering
- Classically, rendering with natural
illumination is very expensive compared to using
simplified illumination
Directional Source
Natural Illumination
64Lighting Invariant Recognition
- Theory Infinite number of light directions
Space of images
infinite-dimensional - Empirical 5D subspace enough for diffuse objects
Images from Debevec et al. 00
65Lighting Invariant Recognition
- Theory Space of images infinite-dimensional
for Lambertian Belhumeur and Kriegman 98
- Empirical 5D subspace enough for diffuse
objects
Hallinan 94, Epstein et al. 95
66Open Questions
- Relationship between spherical harmonics, PCA
- 9D approximation gt 5D empirical subspace
- Key insight Consider approximations over visible
normals (upper hemisphere), not entire sphere
Ramamoorthi CVPR IOAVL 01
67Light Field in 3D
- In flatland, 2D function
- In three dimensions, 4D function
-
Plenoptic Light Field
Surface Light Field
68 Dual Representation
- Diffuse BRDF Filter width small in frequency
domain - Specular Filter width small in spatial (angular)
domain - Practical Representation Dual angular,
frequency-space
69Related Work
- Precomputed (prefiltered) Irradiance maps
Miller and Hoffman 84, Greene 86, Cabral
et al 87 - Empirical observation Irradiance varies slowly
with surface normal. Use low resolution
irradiance maps - Contributions
- Analytic Irradiance formula
- Fast computation
- Compact 9 parameter representation
- Procedural rendering with programmable shading
hardware - Our approach can be extended to general BRDFs
70Comparison
Rendering (known L)
Photograph
Rendering (unknown L)