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Spherical Convolution in Computer Graphics and Vision Ravi Ramamoorthi Columbia Vision and Graphics Center Columbia University SIAM Imaging Science Conference: May 17, 2006

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Outline Motivation and Practical Problems Spherical Convolution and Applications A Theory of Spherical Harmonic Identities Signal Processing for Visual Appearance

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Real-Time Rendering Motivation: Interactive rendering with natural illumination and realistic, measured materials

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Inverse Rendering Photographs Geometric model Inverse Rendering Algorithm BRDF

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Inverse Rendering Geometric model Forward Rendering Algorithm BRDF Novel lighting Rendering Photographs

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Direct Object Relighting ? Unknown Lighting Unknown BRDF

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Lighting-Insensitive Recognition Illuminate subject from many incident directions Space of images as lighting is varied

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Checking Image Consistency Easy to tamper / splice images Image processing software widely available In news reporting and other applications Need to detect tampering or photomontage Verify image consistency Try to check consistency of lighting, shading

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Checking Image Consistency

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Without knowing illumination and reflectance: How do we know if two objects / people in a photograph are lit consistently? How do we detect inconsistencies or tampering?

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Outline Motivation and Practical Problems Spherical Convolution and Applications A Theory of Spherical Harmonic Identities Signal Processing for Visual Appearance

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Environment Maps Miller and Hoffman, 1984 Later, Greene 86, Cabral 87, 99,…

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Irradiance Environment Maps Incident Radiance (Illumination Environment Map) Irradiance Environment Map R N

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Computing Irradiance Classically, hemispherical integral for each pixel Lambertian surface is like a low pass filter Frequency-space analysis (spherical harmonics) Incident Radiance Irradiance

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Spherical Harmonics -2012 0 1 2...2...

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Analytic Irradiance Formula Lambertian surface is low-pass filter 0 01 2 Basri & Jacobs 01 Ramamoorthi & Hanrahan 01a

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9 Parameter Approximation -201 2 0 1 2 Exact image Order 2 9 terms RMS Error = 1% For any illumination, average error < 2% [Basri, Jacobs 01] Ramamoorthi and Hanrahan 01b

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Real-Time Rendering Simple procedural rendering method (no textures) Requires only matrix-vector multiply and dot-product In software or NVIDIA vertex programming hardware Widely used in Games (AMPED for Microsoft Xbox), Movies (Pixar, Framestore CFC, …)

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Computer Vision Complex Illumination Low Dimensional Subspace Lighting Insensitive Recognition (Basri and Jacobs 01, Lee et al. 01, Ramamoorthi 02, …) Photometric stereo, shape acquisition

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Convolution for General Materials Ramamoorthi and Hanrahan 01 Frequency: product Spatial: integral Spherical Harmonics

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Related Theoretical Work Qualitative observation of reflection as convolution: Miller & Hoffman 84, Greene 86, Cabral et al. 87,99 Reflection as frequency-space operator: D’Zmura 91 Lambertian reflection is convolution: Basri Jacobs 01 Our Contributions Explicitly derive frequency-space convolution formula Formal quantitative analysis in general 3D case Apply to real-time, inverse rendering, computer vision

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Natural Lighting, Realistic Materials Ramamoorthi and Hanrahan 02

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Inverse Rendering 3 photographs of cat sculpture Complex unknown illumination Geometry known Estimate microfacet BRDF and distant lighting

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New View, Lighting Photograph Rendering Ramamoorthi and Hanrahan, 01c

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Outline Motivation and Practical Problems Spherical Convolution and Applications A Theory of Spherical Harmonic Identities Signal Processing for Visual Appearance Mahajan, Ramamoorthi, Curless ECCV 06

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Lighting 1 Lighting 2 Material 1Material 2 Two Objects – Two Lightings

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Lighting 1 Lighting 2 Material 1Material 2 Two Objects – Two Lightings

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Lighting 1 Lighting 2 Material 1Material 2 Two Objects – Two Lightings

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Lighting 1 Lighting 2 Material 1Material 2 Two Objects – Two Lightings Independent of Lighting and BRDF

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation Framework ?

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation Framework ?

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation Framework ?

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation – BRDF Transfer ? BRDF Transfer Function

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation – Lighting Transfer ?

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation – Lighting Transfer ?

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Lighting 1 Lighting 2 Material 1Material 2 Image Estimation – Lighting Transfer ? Light Transfer Function

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Image Estimation Object 1 Object 2 Lighting 1 Our MethodActual Lighting 2 ? No BRDF and lighting known or estimated

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Image Consistency Checking Spliced Image

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Image Consistency Checking Spliced Image

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Image Consistency Checking Single Image Identity diffuse + specular case Two Lightings – Same Reflectance Identity Two Materials – Two Lightings identity Tampered Cat Tampered CatUntampered Cat

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Outline Motivation and Practical Problems Spherical Convolution and Applications A Theory of Spherical Harmonic Identities Signal Processing for Visual Appearance

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Signal Processing for Appearance Signal Processing widely applicable visual appearance Convolution relation for cast shadows [Soler and Sillion 98, Ramamoorthi et al. 04] Convolution with glows for participating media (mist, fog, haze) [Sun et al. 05] Signal-Processing analysis of light field and reflectance [Chai et al. 00, Zickler et al. 06] Triple Product Integrals [Ng et al. 04] First Order Analysis [Ramamoorthi et al. 06]

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Video clip 1 Real Time Rendering of Scattering

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Video clip 1 Reflectance Sharing

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Acknowledgements Collaborators Dhruv Mahajan Brian Curless Pat Hanrahan Sameer Agarwal (helpful discussions) Funding: NSF and Sloan Foundation http://www.cs.columbia.edu/~ravir

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