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

2. Radiometry and Image Formation

3. 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!

5. Differential Solid Angle and Spherical Polar Coordinates

7. 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!

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

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

10. 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?

11. 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)

12. 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.

13. 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)

14. 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.

15. The Problem of Dynamic Range

16. 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!!

17. 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.

18. 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.

19. 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.

20. Basic Principles of Surface Reflection

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

22. 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 :

23. 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.

25. Differential Solid Angle and Spherical Polar Coordinates

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

28. 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:

29. 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

30. 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:

31. 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!

32. Diffuse Reflection and Lambertian BRDF

33. 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?

34. 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)

35. 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 :

36. 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)

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

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

39. Components of Surface Reflection

40. 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

41. Dror, Adelson, Wilsky

42. 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 :

43. 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

44. Blurred Highlights and Surface Roughness

45. 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

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

47. 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