Basic Principles of Surface Reflectance Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan

Radiometry and Image Formation

Image Intensities Need to consider light propagation in a cone sensor source Need to consider light propagation in a cone normal surface element Image intensities = f ( normal, surface reflectance, illumination ) Note: Image intensity understanding is an under-constrained problem!

Differential Solid Angle and Spherical Polar Coordinates

Radiometric Concepts (4) Surface Radiance (tricky) : (1) Solid Angle : 2 (4) Surface Radiance (tricky) : (watts / m steradian ) Flux emitted per unit foreshortened area per unit solid angle. L depends on direction Surface can radiate into whole hemisphere. L depends on reflectance properties of surface. source (solid angle subtended by ) (foreshortened area) (surface area) (1) Solid Angle : ( steradian ) What is the solid angle subtended by a hemisphere? (2) Radiant Intensity of Source : ( watts / steradian ) Light Flux (power) emitted per unit solid angle (3) Surface Irradiance : ( watts / m2 ) Light Flux (power) incident per unit surface area. Does not depend on where the light is coming from!

The Fundamental Assumption in Vision Lighting No Change in Radiance Surface Camera

Radiance Properties Radiance is constant as it propagates along ray Derived from conservation of flux Fundamental in Light Transport.

Relationship between Scene and Image Brightness Before light hits the image plane: Scene Radiance L Image Irradiance E Scene Lens Linear Mapping! After light hits the image plane: Image Irradiance E Camera Electronics Measured Pixel Values, I Non-linear Mapping! Can we go from measured pixel value, I, to scene radiance, L?

Relation Between Image Irradiance E and Scene Radiance L image plane surface patch image patch f z Solid angles of the double cone (orange and green): (1) Solid angle subtended by lens: (2)

Relation Between Image Irradiance E and Scene Radiance L image plane surface patch image patch f z Flux received by lens from = Flux projected onto image (3) From (1), (2), and (3): Image irradiance is proportional to Scene Radiance! Small field of view  Effects of 4th power of cosine are small.

Relation between Pixel Values I and Image Irradiance E Camera Electronics Measured Pixel Values, I The camera response function relates image irradiance at the image plane to the measured pixel intensity values. (Grossberg and Nayar)

Radiometric Calibration Important preprocessing step for many vision and graphics algorithms such as photometric stereo, invariants, de-weathering, inverse rendering, image based rendering, etc. Use a color chart with precisely known reflectances. 255 ? Pixel Values g ? 1 90% 59.1% 36.2% 19.8% 9.0% 3.1% Irradiance = const * Reflectance Use more camera exposures to fill up the curve. Method assumes constant lighting on all patches and works best when source is far away (example sunlight). Unique inverse exists because g is monotonic and smooth for all cameras.

The Problem of Dynamic Range

The Problem of Dynamic Range Dynamic Range: Range of brightness values measurable with a camera (Hood 1986) Today’s Cameras: Limited Dynamic Range High Exposure Image Low Exposure Image We need 5-10 million values to store all brightnesses around us. But, typical 8-bit cameras provide only 256 values!!

High Dynamic Range Imaging Capture a lot of images with different exposure settings. Apply radiometric calibration to each camera. Combine the calibrated images (for example, using averaging weighted by exposures). (Mitsunaga) (Debevec) Images taken with a fish-eye lens of the sky show the wide range of brightnesses.

Computer Vision: Building Machines that See Lighting Camera Computer Physical Models Scene Interpretation Scene We need to understand the Geometric and Radiometric relations between the scene and its image.

Computer Graphics: Rendering things that Look Real Lighting Camera Computer Physical Models Scene Generation Scene We need to understand the Geometric and Radiometric relations between the scene and its image.

Basic Principles of Surface Reflection

Surface Appearance sensor source normal surface element Image intensities = f ( normal, surface reflectance, illumination ) Surface Reflection depends on both the viewing and illumination direction.

BRDF: Bidirectional Reflectance Distribution Function source z incident direction viewing direction normal y surface element x Irradiance at Surface in direction Radiance of Surface in direction BRDF :

Important Properties of BRDFs source z incident direction viewing direction normal y surface element x Rotational Symmetry (Isotropy): Appearance does not change when surface is rotated about the normal. BRDF is only a function of 3 variables : Helmholtz Reciprocity: (follows from 2nd Law of Thermodynamics) Appearance does not change when source and viewing directions are swapped.

Differential Solid Angle and Spherical Polar Coordinates

Derivation of the Scene Radiance Equation From the definition of BRDF:

Derivation of the Scene Radiance Equation – Important! From the definition of BRDF: Write Surface Irradiance in terms of Source Radiance: Integrate over entire hemisphere of possible source directions: Convert from solid angle to theta-phi representation:

Mechanisms of Surface Reflection source incident direction surface reflection body reflection surface Surface Reflection: Specular Reflection Glossy Appearance Highlights Dominant for Metals Body Reflection: Diffuse Reflection Matte Appearance Non-Homogeneous Medium Clay, paper, etc Image Intensity = Body Reflection + Surface Reflection

Mechanisms of Surface Reflection Specular Reflection Glossy Appearance Highlights Dominant for Metals Body Reflection: Diffuse Reflection Matte Appearance Non-Homogeneous Medium Clay, paper, etc Many materials exhibit both Reflections:

Diffuse Reflection and Lambertian BRDF source intensity I incident direction normal viewing direction surface element Surface appears equally bright from ALL directions! (independent of ) albedo Lambertian BRDF is simply a constant : Surface Radiance : source intensity Commonly used in Vision and Graphics!

Diffuse Reflection and Lambertian BRDF

White-out Conditions from an Overcast Sky CAN’T perceive the shape of the snow covered terrain! CAN perceive shape in regions lit by the street lamp!! WHY?

Diffuse Reflection from Uniform Sky Assume Lambertian Surface with Albedo = 1 (no absorption) Assume Sky radiance is constant Substituting in above Equation: Radiance of any patch is the same as Sky radiance !! (white-out condition)

Specular Reflection and Mirror BRDF source intensity I specular/mirror direction incident direction normal viewing direction surface element Very smooth surface. All incident light energy reflected in a SINGLE direction. (only when = ) Mirror BRDF is simply a double-delta function : specular albedo Surface Radiance :

BRDFs of Glossy Surfaces Delta Function too harsh a BRDF model (valid only for polished mirrors and metals). Many glossy surfaces show broader highlights in addition to specular reflection. Example Models : Phong Model (no physical basis, but sort of works (empirical)) Torrance Sparrow model (physically based)

Phong Model: An Empirical Approximation An illustration of the angular falloff of highlights: Very commonly used in Computer Graphics

Phong Examples These spheres illustrate the Phong model as lighting direction and nshiny are varied:

Components of Surface Reflection

A Simple Reflection Model - Dichromatic Reflection Observed Image Color = a x Body Color + b x Specular Reflection Color Klinker-Shafer-Kanade 1988 R Color of Source (Specular reflection) Does not specify any specific model for Diffuse/specular reflection G Color of Surface (Diffuse/Body Reflection) B

Dror, Adelson, Wilsky

Specular Reflection and Mirror BRDF - RECALL source intensity I specular/mirror direction incident direction normal viewing direction surface element Very smooth surface. All incident light energy reflected in a SINGLE direction. (only when = ) Mirror BRDF is simply a double-delta function : specular albedo Surface Radiance :

Glossy Surfaces Delta Function too harsh a BRDF model (valid only for highly polished mirrors and metals). Many glossy surfaces show broader highlights in addition to mirror reflection. Surfaces are not perfectly smooth – they show micro-surface geometry (roughness). Example Models : Phong model Torrance Sparrow model

Blurred Highlights and Surface Roughness

Phong Model: An Empirical Approximation How to model the angular falloff of highlights: Phong Model Blinn-Phong Model Sort of works, easy to compute But not physically based (no energy conservation and reciprocity). Very commonly used in computer graphics. N N H R -S E

Phong Examples These spheres illustrate the Phong model as lighting direction and nshiny are varied:

Those Were the Days “In trying to improve the quality of the synthetic images, we do not expect to be able to display the object exactly as it would appear in reality, with texture, overcast shadows, etc. We hope only to display an image that approximates the real object closely enough to provide a certain degree of realism.” – Bui Tuong Phong, 1975