RADIOMETRY (1 lecture) Elements of Radiometry Radiance - PowerPoint PPT Presentation

COS 429: COMPUTER VISON RADIOMETRY (1 lecture) • Elements of Radiometry • Radiance • Irradiance • BRDF • Photometric Stereo Reading: Chapters 4 and 5 Many of the slides in this lecture are courtesy to Prof. J. Ponce

Geometry Viewing Lighting Surface albedo Images I

Photometric stereo example data from: http://www1.cs.columbia.edu/~belhumeur/pub/images/yalefacesB/readme

Image Formation: Radiometry The light source(s) The sensor characteristics The surface normal The surface properties The optics What determines the brightness of an image pixel?

The Illumination and Viewing Hemi-sphere r n ( θ , φ ) ( θ , φ ) At infinitesimal, each point (x, y) i i o o has a tangent plane, and thus a hemisphere Ω . The ray of light is indexed ρ (x, y) by the polar coordinates ( θ , φ )

Foreshortening • Principle: two sources that • Reason: what else can a look the same to a receiver receiver know about a source must have the same effect on but what appears on its input the receiver. hemisphere? (ditto, swapping receiver and source) • Principle: two receivers that look the same to a source • Crucial consequence: a big must receive the same amount source (resp. receiver), of energy. viewed at a glancing angle, must produce (resp. • “look the same” means experience) the same effect as produce the same input a small source (resp. receiver) hemisphere (or output viewed frontally. hemisphere)

Measuring Angle • To define radiance, we require the concept of solid angle • The solid angle sub- tended by an object from a point P is the area of the projection of the object onto the unit sphere centered at P • Measured in steradians , sr • Definition is analogous to projected angle in 2D • If I’m at P, and I look out, solid angle tells me how much of my view is filled with an object

Solid Angle of a Small Patch • Later, it will be important to talk about the solid angle of a small piece of surface θ cos dA ω = d 2 r A ω = θ θ φ sin d d d

DEFINITION: Angles and Solid Angles L = l = θ (radians) R A Ω = a = (steradians) 2 R

DEFINITION: The radiance is the power traveling at some point in a given direction per unit area perpendicular to this direction, per unit solid angle. δ 2 P = L( P, v ) δ A δω δ 2 P = L( P, v ) cos θ δ A δω

PROPERTY: Radiance is constant along straight lines (in vacuum). δ 2 P = L( P, v ) δ A δω δ 2 P = L( P, v ) cos θ δ A δω

DEFINITION: Irradiance The irradiance is the power per unit area incident on a surface. δ 2 P= δ E δ A=L i ( P , v i ) cos θ i δω i δΑ δ E = L i ( P , v i ) cos θ i δω i E= ∫ H L i (P, v i ) cos θ i d ω i

Photometry ⎡ ⎤ 2 π • L is the radiance. ⎛ ⎞ d = α 4 ⎜ ⎟ ⎢ cos ⎥ E L 4 ⎝ ' ⎠ ⎢ ⎥ z • E is the irradiance. ⎣ ⎦

DEFINITION: The Bidirectional Reflectance Distribution Function (BRDF) The BRDF is the ratio of the radiance in the outgoing direction to the incident irradiance (sr -1 ). L o ( P, v o ) = ρ BD ( P, v i , v o ) δ E i ( P, v i ) = ρ BD ( P, v i , v o ) L i ( P, v i ) cos θ i δω i Helmoltz reciprocity law: ρ BD ( P, v i , v o ) = ρ BD ( P, v o , v i )

DEFINITION: Radiosity The radiosity is the total power Leaving a point on a surface per unit area (W * m -2 ). B(P) = ∫ H L o ( P , v o ) cos θ o d ω Important case: L o is independent of v o . B(P) = π L o (P)

DEFINITION: Lambertian (or Matte) Surfaces A Lambertian surface is a surface whose BRDF is independent of the outgoing direction (and by reciprocity of the incoming direction as well). ρ BD ( v i , v o ) = ρ BD = constant. The albedo is ρ d = π ρ BD .

DEFINITION: Specular Surfaces as Perfect or Rough Mirrors θ i θ s θ i θ s δ Perfect mirror Rough mirror Perfect mirror: L o (P, v s ) = L i (P, v i ) Phong (non-physical model): L o (P, v o )= ρ s L i (P, v i ) cos n δ Hybrid model: L o (P, v o )= ρ d ∫ H L i (P, v i ) cos θ i d ω i + ρ s L i (P, v i ) cos n δ

DEFINITION: Point Light Sources N θ i θ i S A point light source is an idealization of an emitting sphere with radius ε at distance R, with ε << R and uniform radiance L e emitted in every direction. For a Lambertian surface, the corresponding radiosity is ⎡ ⎤ ⋅ πε 2 ( ) ( ) N S P P ≈ ρ = ρ θ ( ) ( ) ( ) cos ⎢ ⎥ P B P P L 2 2 d ( ) d e i ( ) ⎣ ⎦ R P R P

Local Shading Model • Assume that the radiosity at a patch is the sum of the radiosities due to light source and sources alone. No interreflections. ⋅ = ∑ ( ) ( ) N S P P ρ • For point sources: ( ) ( ) s B P P 2 d ( ) R P visible s s • For point sources at infinity: ∑ = ρ ⋅ ( ) ( ) ( ) ( ) B P P N P S P d s visible s

Photometric Stereo (Woodham, 1979) ?? Problem: Given n images of an object, taken by a fixed camera under different (known) light sources, reconstruct the object shape.

Photometric Stereo: Example (1) • Assume a Lambertian surface and distant point light sources. I(P) = kB(P) = k ρ N (P) • S = g (P) • V with g (P) = ρ N (P) and V = k S • Given n images, we obtain n linear equations in g : T V 1 . g V 1 I 1 T I 2 V 2 . g V 2 g = V -1 i i = V g i = … = … = g T I n V n . g V n

Photometric Stereo: Example (2) • What about shadows? • Just skip the equations corresponding to zero-intensity pixels. • Only works when there is no ambient illumination.

Photometric Stereo: Example (3) ρ (P) = | g (P) | g (P)= ρ( P ) N (P) 1 = ( ) ( ) N g P P | ( ) | g P

Integrability! Photometric Stereo: Example (3) ⎛ ⎞ ∂ ∂ ∂ ∂ ⎛ ⎞ z z ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ y x x y z ⎡ ⎤ ∂ z − ⎢ ⎥ ∂ ⎧ ∂ y z a ⎡ ⎤ x ⎢ ⎥ = − a ⎪ ∂ ⎪ ⎢ ⎥ ∂ ⎢ ⎥ z x c = ∝ − ⇒ ⎨ N b ⎢ ⎥ ∂ ⎢ ⎥ ∂ z b v y ⎪ = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ c ⎪ ∂ ⎩ y c ⎢ ⎥ 1 u ⎢ ⎥ ⎣ ⎦ x ∂ ∂ z z ∫ u ∫ v = + ( , ) ( , 0 ) ( , ) z u v x dx u y dy ∂ ∂ 0 0 x y

Photometric stereo example data from: http://www1.cs.columbia.edu/~belhumeur/pub/images/yalefacesB/readme